Experimental and Numerical Studies on Microfluidic Systems
Abstract
The work presented in this thesis can be summarized as a compilation of five different and comprehensive studies in the field of microuidic ows, related to the formation of jets, drops and bubbles; where the surface tension plays a major role. The topics studied are classified in chapters where the problem formulation, procedures, and results are individually presented. Chapter 2 is devoted to understand the evolution of Newtonian capillary jets and to study the instability transition of viscoelastic jets under axisymmetric perturbations. A mathematical model has been used to determine the parameter conditions for which the convective to absolute instability transition takes place, playing special attention to the role played by unrelaxed elastic axial stress. Chapter 3 presents results of a numerical study of rivulets in microchannels in order to characterize stable and unstable regimes. The theoretical frame work and stability analysis are presented in detail. It was found that a basic ow can become unstable when that quantity exceeds a certain critical value, while the rest of governing parameters remain constant. Chapter 4 discuses a ubiquitous process in science and technology - the dissolution of microbubbles. As in the previous chapters, detailed theoretical and numerical approaches are developed from scratch, culminating in a set of carefully performed experiments. Numerical and experimental results agree well and complement each other. We move then onto Chapter 5 which studies the electrical disruption of pendant liquid drops. The focus of the study here is the behaviour of suddenly electrified pendant droplets in dielectric liquid. Supported by numerical and experimental results, we argue that the viscosity of the surrounding uid is responsible for the development of more complex jetting processes such as what is called splashing in which the tip of the cone explodes onto a mushroom-like structure, and splitting regimes. Moreover, when the cone evolves into one of these modes they do it in a way that is dependent on the large scale properties such as the initial droplet size and on the applied voltage - contrary to the well-established (universal) mechanisms encountered in the tip streaming mode. Finally, Chapter 6 presents a series of oneto- one numerical and experimental runs with excellent agreement of a novel way of producing drops that are signi_cantly smaller than the nozzle from which they emerge. A very detailed discussion of the experimental rig is presented, including the important parameters to be taken in account, such as the meniscus formed at the nozzle and the deformation of the nozzle plate during the driving pressure pulses. Finally, a predictive scaling law of the produced droplet size was obtained.
Full text
Universidad de Sevilla Escuela T´ecnica Superior de Ingenier´ıa Experimental and Numerical Studies on Microfluidic Systems A dissertation presented by Ahmed Said Mohamed Ismail, to Department of Aerospace Engineering and Fluid Mechanics in fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mechanical Engineering and Industrial Organization. supervised by Dr. Miguel A. Herrada Guti´errez and Dr. J.M. L´opez-Herrera S´anchez Sevilla, April 2016.
Universidad de Sevilla Escuela T´ecnica Superior de Ingenier´ıa Experimental and Numerical Studies on Microfluidic Systems Memoria realizada por Ahmed Said Mohamed Ismail, presentada ante el Departamento de Ingenier´ıa Aerospacial y Mec´anica de Fluidos de la Universidad de Sevilla para optar al grado de Doctor de filosof´ıa en Ingenier´ıa Mec´anica y de Organizaci´on Industrial. Dirigida por Dr. Miguel A. Herrada Guti´errez y Dr. J.M. L´opez-Herrera S´anchez Sevilla, Abril 2016.
Acknowledgements I would like to express my sincere gratitude to my advisors: Dr. Miguel A. Herrada Guti´errez and Dr.J.M. L´opez-Herrera S´anchez for the continuous support of my Ph.D study and related research, for their patience, motivation, immense knowledge and for being there every time I knock their door during these past four years. Also to my tutor Dr. Alfonso M. Ga˜n´an Calvo for giving me the opportunity to be part of his research group, for his guidance, and for many insightful discussions. I learned a lot from them and I hope that our collaboration continue in the future. My sincere thanks also goes to Dr. Joan Rosell-Llompart, and Dr. Alfonso Castrej´on-Pita, who provided me with the opportunity to join their team as intern, and who gave access to the laboratory and research facilities at Universidad Rovira i Virgili and University of Oxford. Without they precious support it would not have been possible to conduct this research. Also, thanks to the Ministry of Economy and Competitiveness of Spain for making this study possible by providing the financial support. My gratitude also goes to my fellow labmates: Luis, Paco and Irene for sharing thoughts and for having interesting discussions. I thank also Beatriz and Dory for their help in solving Administrative problems. My thanks goes also to my family: my brother and my sisters and especially to my parents for being such a big source of supporting and encouragement without both of you I couldn’t be here. Finally, thanks to my beloved wife for supporting me spiritually. i
Abstract The work presented in this thesis can be summarized as a compilation of five different and comprehensive studies in the field of microfluidic flows, related to the formation of jets, drops and bubbles; where the surface tension plays a major role. The topics studied are classified in chapters where the problem formulation, procedures, and results are individually presented. Chapter 2 is devoted to understand the evolution of Newtonian capillary jets and to study the instability transition of viscoelastic jets under axisymmetric perturbations. A mathematical model has been used to determine the parameter conditions for which the convective to absolute instability transition takes place, playing special attention to the role played by unrelaxed elastic axial stress. Chapter 3 presents results of a numerical study of rivulets in microchannels in order to characterize stable and unstable regimes. The theoretical frame work and stability analysis are presented in detail. It was found that a basic flow can become unstable when that quantity exceeds a certain critical value, while the rest of governing parameters remain constant. Chapter 4 discuses a ubiquitous process in science and technology - the dissolution of microbubbles. As in the previous chapters, detailed theoretical and numerical approaches are developed from scratch, culminating in a set of carefully performed experiments. Numerical and experimental results agree well and complement each other. We move then onto Chapter 5 which studies the electrical disruption of pendant liquid drops. The focus of the study here is the behaviour of suddenly electrified pendant droplets in dielectric liquid. Supported by numerical and experimental results, we argue that the viscosity of the surrounding fluid is responsible for the development of more complex jetting processes such as what is called splashing in which the tip of the cone explodes onto a mushroom-like structure, and splitting regimes. Moreover, when the cone evolves into one of these modes they do it in a way that is dependent on the large scale properties such as the initial droplet size and on the applied voltage - contrary to the well-established (universal) mechanisms encountered in the tip streaming mode. Finally, Chapter 6 presents a series of oneto-one numerical and experimental runs with excellent agreement of a novel way of producing drops that are significantly smaller than the nozzle from which they ii
emerge. A very detailed discussion of the experimental rig is presented, including the important parameters to be taken in account, such as the meniscus formed at the nozzle and the deformation of the nozzle plate during the driving pressure pulses. Finally, a predictive scaling law of the produced droplet size was obtained. iii
Contents 1 Introduction 1 1.1 Production and dissolution of microbubbles . . . . . . . . . . . . . . 3 1.2 Production of microdroplets . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Hydrodynamic jetting . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Electrohydrodynamic jetting . . . . . . . . . . . . . . . . . . . 9 1.2.3 Other methods to produce droplets . . . . . . . . . . . . . . . 12 1.3 Formation of capillary jet . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Objectives and document structure . . . . . . . . . . . . . . . . . . . 15 2 Instability transition in a viscoelastic capillary jet 17 2.1 Introduction................................ 17 2.2 Mathematicalmodel ........................... 18 2.3 Governing equations and numerical method . . . . . . . . . . . . . . 21 2.4 Results................................... 23 2.5 Conclusions ................................ 28 3 Stability of a rivulet in a microchannel 31 3.1 Introduction................................ 31 3.1.1 The governing equations . . . . . . . . . . . . . . . . . . . . . 35 3.1.2 Basicflow ............................. 38 3.1.3 Linear perturbations . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.4 Temporal stability analysis . . . . . . . . . . . . . . . . . . . . 42 3.1.5 Results............................... 42 3.2 Conclusions ................................ 48 4 Dissolution of Micro-bubbles 50 4.1 Introduction................................ 50 4.2 Problemformulation ........................... 54 4.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.2 Numerical procedure . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Experimentalsetup............................ 66 iv
4.3.1 Apparatus Configuration . . . . . . . . . . . . . . . . . . . . . 66 4.3.2 Objective and Procedures . . . . . . . . . . . . . . . . . . . . 69 4.3.3 Diffusivity, Henry’s constant and the Partial pressure . . . . . 70 4.4 Results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 Electrical disruption of pendant liquid drops 77 5.1 Introduction................................ 77 5.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . 79 5.2.1 Numericalmodel ......................... 80 5.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3.1 Subcritical regime . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3.2 Supercritical regime: The splashing mode ........... 91 5.4 Conclusions ................................ 97 6 Small drops from large nozzles 99 6.1 Introduction................................ 99 6.2 Experimentalsetup............................102 6.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4.1 Experimental jetting process . . . . . . . . . . . . . . . . . . . 107 6.4.2 Numerical to experimental results . . . . . . . . . . . . . . . . 110 6.4.3 Scalinganalysis..........................113 6.5 Conclusions ................................115 List of Figures 117 List of Tables 122 Bibliography 123 v
Chapter 1 Introduction Any review of the literature on the microfluidic systems related to capillary jets, droplets and bubbles will reveal in short their interest and relevance. For instance, a recent quick search in the SCOPUS database has returned 39605 items when the word searched is “microfluidic”. If this search is combined with ”jet”, ”droplet” or ”bubble” the figures of found items is still enormous; 1568, 8132 and 3253, respectively. These large figures are just a reflect of the great importance in many technological applications of such systems. Capillary jets can be found in an enormous variety of processes of interest in very diverse fields, such as pharmacy, biotechnology, industrial and chemical engineering, or the food and agriculture industry. Typically in these fields the objective is to use the capillary threads as a mid-stage step on the production of solid fibers. A variety of physico-chemistry processes can be used to solidify the produced liquid threads before their breakup. In this way, that sub-millimeter artificial fibers for the textile industry can be obtained[1]. Note that the world production of textile fibers alone has increased from about 24 million metric tons in 1975 to nearly 90 million tons in 1
2 1. Introduction 2014, and the percentage of artificial fibers rose from around 50% in 1975 to nearly 75% in 2014. Formation of drops is involved in many applications as for example, DNA arraying, the printing of electronics and biomaterials, drug delivery, inkjet printing, automatic pipetting of fluids and manufacture of particles. While, microbubbles are used in drug delivery, biofilm removal and membrane cleaning, also they are used as a contrast medium in images in medical diagnostics. In this thesis, experimental and numerical studies have been carried out in order to investigate diverse aspects of jets, drops and bubbles in micro/nano scale. Since these fields are too broad, we have limited the aim of the present thesis to, 1. The study of viscoelastic jets. In particular we will perform the spatiotemporal stability of viscoelastic jets subject to unrelaxed axial stress. 2. The study of the capillary instability of rivulets in order to control the production of fibers and microdroplets/microbubbles, respectively, by performing stability analysis of base (zeroth) solutions. 3. The study of the dissolution of gaseous bubbles ascending through a liquid media. The analysis is either numerical and experimental. 4. The study of the liquid-liquid electrical dispersion problem. This issue was also studied experimentally and numerically. 5. The study of “drop on demand” generator based in a novel collapsing technique. Note that the first item could be categorized as related to capillary jets while the rest falls in the generic field of droplet systems except item three which is related to
1.2. Production of microdroplets 9 GLR =mg/ml,(1.6) being ρg, ρl, Ug, D, mg, and mlare gas density, liquid density, gas velocity, nozzle diameter, mass flow rate of gas, and mass flow rate of liquid respectively. Figure 1.3: Configuration difference between FF an FB (Ga˜n´an-Calvo, 2005). 1.2.2 Electrohydrodynamic jetting One important method to produce very small and monodisperse droplets is the conejet electrospraying (ES) which is known also as Taylor’s cone [24, 25]. The idea of this method is to apply an electric field to liquid meniscus. Then charges accumulates on the interface producing Maxwell stresses which is proportional to the permittivity of vacuum. The meniscus stretches and forms a conical shape when a balance occur between pressure drop across the interface, the surface tension and Maxwell
10 1. Introduction stresses. Subsequently, a microjet is emitted from its tip for a voltage difference above the critical (see fig 1.4 (a)). The microjet breaks up downstream producing the droplets. Voltage difference is the energy source in ES which is analog to pressure drop in FF. In electrohydrodynamic jetting, another dimensionless numbers appear, which are electric Bond number, relative permittivity, and dimensionless electrical conductivity as: Be=εiE2R/σ, (1.7) β=εi/εo,(1.8) Kd=K(ρR3/σε2 o)1/2,(1.9) being Kand εiare conductivity and permittivity of liquid, Eis the electric field, and εois the permittivity of the ambient. + _ (A. Barrero et al., 2003) Figure 1.4: (a) Electospraying in air ambient, (b)Electrospraying in liquid ambient.
1.2. Production of microdroplets 11 The droplets produced are charged so, electrospraying has been applied widely in mass spectrometry of large biomolecules. Electrohydrodynamic jetting is not only useful to spray liquids in gas ambient but also into another liquid (see fig 1.4 (b)). Liquid-Liquid dispersion is an essential step in many industrial processes nowadays. It is part, for example, of applications like the encapsulation of drugs or food additives or the obtention of micro(nanometric)-range emulsions. Electrical forces have been proven as an efficient dispersion mechanism. Watanabe et al. [26] produced water-in-oil emulsions by applying a potential difference higher than the critical voltage of emulsification. Their method was based on counteract surface tension forces by means of electrical forces. They introduced a large quantity of additives in the dispersed phase so that the electrical conductivity of continuous-phase liquid were smaller than that of the dispersed-phase liquid. Sato and coworkers[27] used the Inversed Electrostatic Spraying (IES) to disperse a dielectric liquid in a conducting liquid medium. The IES is the technique introduced by Tsouris et al. [28] to disperse a gas into an outer conducting medium by means of electrical fields. In this technique the dispersed phase is injected continuously through an electrified, in most cases metallic, needle. A grounded electrode is located downstream in the bath chamber in order to create an intense enough electric field. Some improvements of this technique can be found in the literature. More intense electric fields (resulting in smaller droplets) can be obtained by covering the metallic capillary nozzle with an insulating material up to the needle’s tip[29]. Tsouris et al.[30] illustrated that a conical tip sharpened capillary and negative polarity provide better pumping, spraying and mixing. Gneist & Bart[31] used a high frequency AC power supply. There was no effect of viscosity of dispersed phase on spraying process for viscosities up to 100 mPa.s. In most of the aforementioned examples the droplet dispersion
12 1. Introduction occurs within the electrified dripping regime[32]. Experimental and numerical study was conducted in chapter 5 to explore new modes of electrohydrodynamic jetting in liquid-liquid dispersion. 1.2.3 Other methods to produce droplets Another method called drop-on-demand (DOD) ink jet printing, to produce drops, is used widely in applications like printing [33], fabrication of transistors [34] and biochip arraying [35]. There are two DOD technologies that predominate in ink jet industry: piezo and thermal ink jet (TIJ). Piezo technology was developed by Zoltan in 1972 [36] and by Kyser and Sears in 1976 [37]. In this technology, a capillary tube made of glass is bonded to piezoelectric transducer (see fig. 1.5 (a)). The transducer receives an electric signal to produce pressure wave. The wave squeezes or relaxes the tube so, a drop of the same order of the tube size can be ejected from the nozzle by selecting the suitable voltage pulse. The great deal of this method is to get the correct shape of the waveform, amplitude and duration of the voltage pulse. Thermal ink jet (TIJ) was developed separately by Canon [38] and Hewlett-Packard [39]. The idea of this method is to use the expansion of vapor bubble formed on a heating element near to the nozzle exit in order to eject a drop from the nozzle (see fig. 1.5 (b)). One of the most important goals of many scientist is to reduce the volume of the generated drop, in order to increase the resolution in printing and to reduce the consumption of ink. So, we have conducted numerical and experimental study en chapter 6 to have a better control on a novel technique to produce drop on demand.
1.3. Formation of capillary jet 13 Figure 1.5: Piezo and thermal ink jet technologies, (Basaran, 2002). 1.3 Formation of capillary jet Capillary jets are produced by exerting intense axial forces to overcome the resistance offered by viscosity and surface tension. In the simplest example of ejection through a nozzle or spinneret, a strong pressure drop must be applied to the feeding capillary and the ambient to provide the liquid with kinetic energy sufficient to create the interface. In flow-focusing and electrospray (electrospinning) [40], an outer stream and electric field, respectively, stretch a liquid meniscus until a thin jet tapers from its very tip. In the case of viscoelastic jets, an intense axial elastic tension is generated in the ligament at the ejection point. Viscoelastic material exhibits both viscous and elastic behaviour when a force is applied to it. So, it forms a longer jet than other fluids. If no further stretching occurs, the tension decays at a rate that decreases as elasticity increases [41, 42, 43]. For high enough elasticity, the axial tension survives downstream over distances much longer than the jet’s diameter. Most experiments with viscoelastic jets are conducted for a relatively high elasticity,
14 1. Introduction so that its effects can be clearly appreciated. In this case, the axial tension becomes a fundamental term in the analysis of the stability of non-Newtonian jets. If the unperturbed state is subject to unrelaxed axial elastic tension, then the nonlinear terms of the rheological model contribute to the linearized equations, and therefore the results depend on the specific model even in the linear regime. The Oldroyd-B constitutive equation [44] provides reasonably accurate predictions for viscoelastic liquids under certain conditions. Specifically, it can describe both the linear and nonlinear evolution of the so-called Boger liquids [45]. These liquids are dilute polymer solutions in solvents with a sufficiently high viscosity for elastic stresses to be measurable. They exhibit a constant viscosity (shear thinning can be neglected) over a wide range of shear rates, so that the elastic effects can be separated from the viscous ones. Goren and Gottlieb [41] analyzed the temporal linear stability of the axisymmetric mode in an Oldroyd-B capillary jet. They showed that the growth rate decreases as the axial tension increases, although there is always a range of wave numbers for which the mode is unstable. Ruo et al. [42] extended that analysis to non-axisymmetric perturbations in the presence of an unbounded inviscid gas. They also concluded that the axial stress plays a stabilizing role (these results must be reviewed because the effects of the unrelaxed tension were not accounted for correctly in the boundary conditions at the free surface). The stability analysis of annular liquid sheets also leads to the same conclusion [46]. Neither the temporal nor the spatial stability analysis can predict whether steady jetting will occur in an experimental realization. For that purpose, the convectiveto-absolute instability transition must be determined from a spatiotemporal analysis [47]. If the jet is convectively unstable, an unperturbed cylindrical ligament forms next to the discharge orifice, while growing surface waves deform and eventually
1.4. Objectives and document structure 15 pinch the interface downstream. On the contrary, absolute instability implies that perturbations travel both downstream and upstream over the jet’s surface, preventing steady jetting. The convective-to-absolute instability transition has been analyzed in relaxed capillary jets exclusively [48]. In line with the temporal analysis results [41, 42, 46], one concludes that elasticity plays a destabilizing role, fostering absolute instability at the expense of the convective one. This result was subsequently extended to electrified jets [49]. 1.4 Objectives and document structure Through this thesis we pretend to study some problems aspects to produce fibers, drops and bubbles. Our objective is to study all the parameters that govern those problems in order to control the production process in efficient manner. In chapter 2, a numerical model has been developed to do a stability analysis for an Oldroyd-B capillary jet subject to unrelaxed axial stress. Our purpose is to determine the convective to absolute instability transition. By this mean, a better control can be made on the production of fibers of viscoelastic materials. In chapter 3, stability analysis of rivulet in mirochannel has been realized by using numerical model. The objective is to characterize the stable and unstable regimes. This study helps us to get a better knowledge of the limits where we can produce drops or bubbles using T-junction technique. In chapter 4, our target is to obtain a reliable model to predict the evolution of a bubble of gas that slowly dissolves in an open environment, and whose size is sufficiently small to remain spherical –even subject to its buoyant rise through the surrounding liquid. In particular, our interest focuses on the total dissolution
16 1. Introduction time and distance traveled by the evolving microbubble, with the idea to make that distance traveled as close as possible to the distance from the bubble source to the free surface of the liquid: in principle, this would allow an optimal dispersion of the gas throughout the liquid column, with minimal gas loses at the free liquid surface. In chapter 5, the objective is to study all parameters that govern the problem of the electrical disruption of pendant liquid drops in another liquid. For this purpose, a broad series of experiments have been done. Also, a numerical code was used in this study. The results reveal new modes in liquid-liquid dispersion and determine the limits of the production of micro-droplets in another liquid. In chapter 6, experimental and numerical studies have been achieved on a novel technique to produce drop on demand. The target is to control the different parameters affecting on the production process and to use the experimental and numerical results to obtain a predictive scaling law for the droplet size. The results in chapter 2, 3 and 4 have been published in the following articles: •A. Said Mohamed, M. A. Herrada, and A. M. Ga˜n´an-Calvo and J. M. Montanero, ”Convective-to-absolute instability transition in a viscoelastic capillary jet subject to unrelaxed axial elastic tension” Physical Review E, 92, 023006, August 2015. •A. Said Mohamed, Miguel A. Herrada, J.M. L´opez-Herrera and Alfonso M. Ga˜n´an-Calvo, ”Isothermal dissolution of small rising bubbles in a low viscosity liquid” Chemical Engineering and Processing, 85:136-144, November 2014. •Miguel A. Herrada, A.S. Mohamed, Jos´e M. Montanero and A.M. Ga˜n´anCalvo ”Stability of a rivulet flowing in a microchannel” International Journal of Multiphase Flow, 69:1-7, October 2014.
Chapter 2 Instability transition in a viscoelastic capillary jet 2.1 Introduction Many practical applications involve the formation and controlled breakup of viscoelastic laminar jets. Elasticity alters fundamentally the evolution of the capillary jet not only in the ultimate nonlinear stage of its breakup, but also in the initial linear regime. It is well-known that the growth rates characterizing the linear instability of a relaxed (zero axial elastic tension) capillary jet are greater than their counterparts in the Newtonian case [50, 51, 52]. Therefore, elasticity plays a destabilizing role for vanishing axial tension in the unperturbed state. Interestingly, these predictions are independent of the form of the viscoelastic constitutive equation. If the axial tension of the base solution vanishes, all nonlinear terms in the constitutive relationship are eliminated after linearization, which leads to the general linear viscoelastic Jeffreys model [52]. 17
18 2. Instability transition in a viscoelastic capillary jet The theoretical results mentioned above clearly contrast with most experimental observations, which repeatedly show that elasticity stabilizes the produced jets, delaying considerably their breakup [53] (a feature exploited in the industrial production of polymeric fibers). It has been argued that this discrepancy is due to the inability of the linear stability analysis to provide relevant information about the jet disintegration process. Although it is obvious that linear predictions cannot be extrapolated beyond the very first stage of the thread breakup, there is yet another effect that may also explain the lack of agreement between the linear stability predictions and the experiments: the jet’s axial elastic tension [41, 42, 46]. In this chapter, we will conduct a spatiotemporal linear stability analysis of the Navier-Stokes equations for an Oldroyd-B capillary jet subject to unrelaxed axial stress. We will determine the parameter conditions for which the convectiveto-absolute instability transition takes place, paying special attention to the role played by the unrelaxed elastic axial stress. 2.2 Mathematical model In this section, we present the mathematical model with abbreviated expressions, while the expanded equations can be found in the next section. Consider a cylindrical jet of radius Rand density ρmoving in the axial direction at the uniform velocity V. The properties of the outer medium are such that its dynamical effects on the jet can be neglected. Due to the smallness of the jet, the gravity effects are neglected too. In what follows, we shall make all the variables dimensionless using the radius R, velocity V, convective time R/V , and dynamic pressure ρV 2as the characteristic length, velocity, time, and pressure, respectively. The jet rheological behavior is
2.4. Results 25 restrict ourselves to the case β= 0 because this parameter takes very small values in most experiments with polymeric solutions. Figure 2.2 shows the critical Weber number as a function of the Reynolds number for two values of the Deborah number, λ= 1 and 100, and three values of the unrelaxed axial stress, τ0= 0,0.1, and 10. The numerical results perfectly match the analytical solution [48] for τ0= 0 and λ= 100, which shows the accuracy of our numerical approach. The figure also shows the convective-to-absolute instability transition curve for a Newtonian jet (λ=τ0= 0) [59]. As showed by by Montanero and Ga˜n´an-Calvo [48], elasticity enhances the absolute instability in a relaxed (τ0= 0) capillary jet over most part of the analyzed interval of the Reynolds number. The unrelaxed tension only stabilizes the liquid thread for sufficiently large Deborah and Reynolds numbers. It must be noted that the limit Re →0 must be taken with caution, because the effects of the outer medium cannot be neglected in that case even for very small density and viscosity ratios. The stabilizing effect of the unrelaxed tension can be clearly appreciated in Fig. 2.3, where the dependency of the critical Weber number upon ˆτ0=τ0/We is shown. This latter quantity is the unrelaxed axial stress in terms of the dynamic pressure ρV 2(instead of the capillary pressure γ/R). Contrary to what one might expect from the temporal analysis, the critical Weber number increases with ˆτ0for λ= 1. In this case, the axial stress favors the upstream climbing of unstable capillary waves (absolute instability), while reducing their growth factors. For λ= 10 and 100, the transitional Weber number decreases as the unrelaxed stress increases until a turning point is reached. That point results from the crossover of two solution branches. Only the dominant branch (i.e., that with the highest Weber number) is plotted in the figure. This crossover corresponds to the existence of a double pinching
26 2. Instability transition in a viscoelastic capillary jet Figure 2.3: Critical Weber number We∗as a function of the unrelaxed axial tension ˆτ0for Re=0.1,1, and 100, and λ= 1,10, and 100. verifying Brigg’s condition, a phenomenon also observed in compound capillary jets [60]. Figure 2.4 illustrates this peculiar situation for the case {We = 0.164, Re=1, λ= 10, τ0= 1.04}, where one can observe how two saddle points “couple each other” through one of the spatial branches. This circumstance has a purely mathematical character, because it corresponds to a convective-to-absolute transition that could not be distinguished from others experimentally. The dominant solution branch for large axial stresses leads to a threshold of this
2.4. Results 27 Figure 2.4: Double pinching with zero growth rate for We=0.164, Re=1, λ= 10, and τ0= 1.04. The symbols correspond to spatial branches with ωi= 0. The arrows indicate the direction in which ωrincreases. quantity above which the viscoelastic jet becomes absolutely unstable independently of the Weber number. This critical value is ˆτ0≃4.63 for sufficiently large Reynolds and/or Deborah numbers. This conclusion can also be drawn from Fig. 2.5, where the curves already plotted in Fig. 2.3 have been organized in a different manner. The existence of such a critical unrelaxed stress can be interpreted in the following way. The speed at which growing waves travel over unrelaxed viscoelastic jets increases with the elastic axial stress [61]. One may expect that if this stress, measured in terms of the dynamic (convective) pressure, exceeds a certain threshold, then the jet will fail to sweep downstream those waves (absolute instability). Therefore, the reduction of both the growth rates and the critical Weber numbers due to the unrelaxed stress is not contradictory but reasonable.
28 2. Instability transition in a viscoelastic capillary jet Figure 2.5: Critical Weber number We∗as a function of ˆτ0for λ= 1,10 and 100, and Re=0.1,1 and 100. 2.5 Conclusions To conclude, we have examined the convective-to-absolute instability transition under axisymmetric perturbations in an Olroyd-B capillary jet subject to unrelaxed axial stress. There is a critical Weber number below which the jet becomes absolutely unstable. The unrelaxed stress destabilizes the viscoelastic jet for small values of the Deborah number. For higher values of this parameter a more complex scenario
2.5. Conclusions 29 arises. The transitional Weber number decreases as the unrelaxed stress increases until two solution branches cross each other. The dominant branch for large axial stresses yields a threshold value of this quantity above which the viscoelastic jet becomes absolutely unstable independently of the Weber number. This threshold takes a universal value for sufficiently large Reynolds and Deborah numbers. It is well-known that the unrelaxed axial stress in a viscoelastic jet increases the speed at which capillary waves move over the jet’s surface [61]. This effect allows one to understand why the axial stress may favor absolute instability while reducing the growth rates. The linear stability analysis for Newtonian capillary jets provides valuable predictions that can be extended to the nonlinear regime. For this reason, the convectiveto-absolute instability transition calculated with the linearized Navier-Stokes equations has been successfully linked to the jetting-to-dripping transition (see, e.g., [62, 63, 64]). On the contrary, polymers are significantly stretched by the growth of axisymmetric perturbations in a viscoelastic jet, which alters drastically the nonlinear behavior of this system [53]. A natural question is whether the convectiveto-absolute instability transition for linear perturbations in a viscoelastic capillary jet corresponds to a true jetting-to-dripping transition. In fact, one may expect the absolute instability to manifest itself as sustained oscillations over the viscoelastic jet, rather than as the appearance of a dripping-like, beads-on-a-string, or blistering mode. Most Boger liquids are manufactured by dissolving polymer solutes in water, and thus the capillary jets are characterized by Reynolds numbers (based on the solvent viscosity) on the order of or greater than unity. Previous results for Newtonian liquids indicate that the effect of an outer gaseous medium can be neglected except
30 2. Instability transition in a viscoelastic capillary jet in the limit of vanishing Reynolds numbers. Therefore, this approximation is also expected to hold in most experimental realizations with viscoelastic liquids.
Chapter 3 Stability of a rivulet in a microchannel 3.1 Introduction When a jet touches and sticks to a solid wall, it forms a rivulet. This fluid configuration plays an important role in a number of industrial applications. Here, we just mention some examples. The interfacial shear caused by the overlying gas in heat exchangers significantly affects the performance of these devices. Rivulets driven by the shear force exerted by the surrounding air are frequently considered when studying the icing of aircraft components. When gravity is the driving force, the rivulet flow is exploited in trickle bed reactors and structured packings. Rivulets are also formed to producing coated surfaces for varied applications. In this case, one may be interested either in the formation of very uniform coatings or in the generation of certain fluid patterns. While in the former case the aim is to quench the instability mechanisms, unstable shapes are exploited in the latter one to producing 31
32 3. Stability of a rivulet in a microchannel the desired pattern. Owing to the Rayleigh capillary instability [65], Newtonian laminar jets break up into streams of droplets whose diameters commensurate with that of the precursor jet. In this way, relatively monodisperse collections of droplets can be obtained from jetting realizations to build, for instance, functional materials for health care and pharmacy [66]. On the other hand, polydisperse sprays can be produced from the breakage of liquid ligaments when turbulence sets in [67]. Fluid rivulets can be produced in microfluidic devices by printing micrometer hydrophilic/hydrophobic stripes on the channel surface. A number of methods can be used for this purpose, including vapor deposition through grids, elastomer stamps, domain formation in Langmuir-Blodgett monolayers, and photolithography of amphiphilic monolayers. These chemical ducts can only be created if the contact angles characterizing the lyophilic and lyophobic surfaces verify certain conditions [68]. The use of chemical ducts in microfluidics prevents from clogging by solute particles, such as colloids or large bio-polymers, which constitutes an important advantage. Based on this idea, Herrada et al. [10] have recently proposed a microfluidic technique to produce quasi-monodisperse collections of microbubbles in a controlled manner. In this technique, a gaseous stream is injected through a T-junction into a channel transporting a liquid current. A hydrophobic strip is printed on one of the channel surfaces, and thus the gas stream forms a rivulet over that strip. If the rivulet is convectively unstable [47], it breaks up downstream due to a capillary pearling instability, which leads to a quasi-monodisperse collection of microbubbles that can be much smaller than the channel size. For the sake of illustration, Fig. 3.1 shows numerical simulations of the fluid configuration analyzed by Herrada et al. [10]. Image (a) corresponds to the case in which the gaseous rivulet does not form, while images (b) and (c) show a convectively unstable and stable rivulet, respectively.
3.1. Introduction 33 Figure 3.1: Flow snapshots for an air-ethanol rivulet [10]. The images correspond to three different regimes depending on the gas Qgand liquid Qlflow rates: (a) bubbling (Qg= 3.6 ml/h and Ql= 36 ml/h), (b) convectively unstable rivulet (Qg= 1.8 ml/h and Ql= 72 ml/h), and (c) stable rivulet (Qg= 0.09 ml/h and Ql= 72 ml/h). The linear stability of liquid rivulets has been frequently studied over the last two decades. It crucially depends on the behavior of the triple contact lines [69]. If they are allowed to move, the rivulet suffers from pearling instability in any case (analogously to what happens in jets). If they are perfectly pinned, then a static rivulet is unconditionally stable for an unperturbed contact angle lower than 90◦, while there is a range of unstable wavenumbers if the contact angle exceeds that threshold. These results were originally obtained for an infinite rivulet resting on a flat surface [70, 71], and subsequently extended to more complex equilibrium configurations, such as rivulets of finite length [72, 68] or rivulets lying on substrates of varied shapes [73]. The stability of flowing rivulets with anchored contact lines has also been considered by several authors. The lubrication and thin-film approximations have been
34 3. Stability of a rivulet in a microchannel used to calculate the basic flows driven not only by the gravitational force [74, 75] but also by other factors, such as a prescribed uniform transverse shear stress at its free surface [76]. Under those approximations, rivulets flowing over a vertical plane [77] and under a sloping plate [78] have proved to be stable as long as their triple contact lines are fixed. Weiland and Davis [79] have shown that shallow rivulets with pinned contact lines flowing down over a vertical surface become unstable if the driving force exceeds a critical value which increases as the Reynolds number decreases. Koplik et al. [80] studied the stability of nano-rivulets driven by gravity from both the linear stability analysis of the Navier-Stokes equations and molecular dynamics simulations. The configurations analyzed were stable if and only if the contact angle was lower than 90◦. Linear stability analysis provides quantitative predictions for the size of the droplets resulting from the rivulet breakup. Diez et al. [81] have found that the distance between the drops that form during the nonlinear evolution is essentially determined by the wavelengths predicted by the linear approximation. Herrada et al. [10] have also shown a good agreement between the droplet size obtained from the linear stability analysis and the simulations of the full Naver-Stokes equations. Several authors have studied the rivulet’s stability by determining when it is energetically favorable for the rivulet to break up into sub-rivulets. Schmuki and Laso [82] found that this is the case for thin rivulets on a sloping plate under certain conditions. The combined action of a body force and a uniform longitudinal shear stress have been examined from the energy approach too [83, 84, 85, 86]. For instance, Wilson and Duffy [85] determined the conditions for which a thin rivulet on a inclined substrate and in the presence of a prescribed uniform longitudinal shear stress splits into sub-rivulets. It must be noted that it is not clear how lin-
3.1. Introduction 41 and the stresses on both sides of the interface are balanced, τ(1) t1=µτ(2) t1, τ(1) t2=µτ(2) t2, p(1) −p(2) −Re−1(τ(1) n−µτ(2) n) = We−1∇·n∗,(3.22) τ(j) t1=1 F3 0a2a1v(j)−2F0F0 0v(j) θ−a1F0v(j) r+ 2F2 0F0 0u(j) r−a1u(j) θ −2F0F0 0u−ikfaa3),(3.23) τ(j) t2=1 F4 0a3−F0 0a2w(j) θ+a2F2 0w(j) r+F2 0a2iku(j)−F0F0 0a2ikv(j) +fa4−f0a3),(3.24) τ(j) n=2 F3 0a2F0F0 0v(j)+F02 0v(j) θ−F2 0F0 0v(j) r+F02 0u(j)−F0 0F0u(j) θ +F3 0u(j) r+ikafa5,(3.25) a1= (F02 0−F2 0), a2= (F02 0+F2 0), a3=F3 0(W0θ+F0 0W0r),(3.26) a4=a2F0 0W0θ−F0F0 0W0rθ +F3 0W0rr, a5=F2 0F0 0W0θ−F2 0W0r,(3.27) ∇·n∗=−f F0aa22F02 0+F2 0−a2F2 0k2−F0F00 0−f0 aa2 2F0F3 0F0 0+ 4F03 0F0 −3F2 0F0 0F00 0−f00F0 aa2 .(3.28) At the solid walls and the edge of the hydrophilic/hydrophobic strip, no-slip u(j)= v(j)=w(j)= 0 and anchorage f= 0 conditions are imposed, respectively. Finally, all the fields are symmetric functions with respect to the mid-plane θ=π/2.
42 3. Stability of a rivulet in a microchannel 3.1.4 Temporal stability analysis The solution of the temporal linear stability analysis is obtained in terms of the set of parameters {L,θs0,ρ,µ,Re,We}. First, the basic flow is calculated by solving the Poisson equations (3.10) with the corresponding boundary conditions. This solution is used in the linear problem (3.16)-(3.22) to get a homogeneous system of equations whose solvability condition leads to the dispersion relation D(k, Ω) = 0. Both the basic flow and linear perturbations are calculated numerically. The rivulet and coflowing stream domains are mapped onto fixed quadrangular domains through a coordinate transformation. The equations are discretized in the radial direction by expanding the fields in terms of truncated Chebyshev series [55, 60]. The angular derivatives were calculated with fourth-order central finite differences using uniformly distributed points. Both the fluid domain mapping and the spectral discretization accumulate the grid points in the vicinity of the free surface and solid walls, where larger gradients of the hydrodynamic fields are expected. This discretization method allows one to get very accurate results with a reduced number of grid points. Use is made of the Matlab subroutine Eigs to calculate the eigenvalues corresponding of the resulting system of linear equations. 3.1.5 Results In this section, we present the values of the growth factor Ωiobtained for two fluid configurations: an air rivulet surrounded by ethanol (ρ= 650 and µ= 83.8) and an ethanol rivulet surrounded by air (ρ= 1.54 ×10−3and µ= 1.19 ×10−2). We considered L= 10 (a value close to that of the experimental analysis that is currently underway) and three values of the unperturbed contact angle: θs0= 80◦, 89◦, and
3.1. Introduction 43 Figure 3.3: Growth factor Ωifor an air rivulet surrounded by ethanol with Re = 3, We = 0.04, and θs0= 89◦(left) and 120◦(right). 120◦. The Weber number was We = 0.04, 1, and 5, while the Reynolds number was varied over several orders of magnitude. Figure 3.3 illustrates the dependence of the growth factor Ωiwith respect to the wavenumber for two air rivulets. In both cases, two capillary modes can be found within the range of wavenumbers and growth factors analyzed. The dominant mode, i.e., that for which the growth factor takes the maximum value, Ωmax i, determines the system stability. If Ωmax i<0, then the rivulet is stable under the perturbations considered, while it becomes unstable otherwise. As can be observed, the rivulet with θs0= 89◦(left-hand graph) is stable because all the perturbations are damped by viscous stresses. On the contrary, the case θs0= 120◦(right-hand graph) corresponds to an unstable configuration because the growth factor characterizing one of the modes becomes positive as θs0increases, and thus that mode grows driven by the surface tension force. We have verified that Ωr6= 0 in all the cases analyzed, which means that the perturbation possesses an oscillating character. Figure 3.4 shows the isolines of the magnitude of the velocity field perturbation
44 3. Stability of a rivulet in a microchannel Figure 3.4: Isolines of the magnitude of the velocity field perturbation corresponding to the unstable (left) and stable (right) capillary modes obtained for k= 0.4 in an air rivulet surrounded by ethanol with Re = 3, We = 0.04, and θs0= 120◦. The red (blue) lines correspond to the higher (lower) values. The results are normalized with the maximum value in each case. corresponding to the two capillary modes with k= 0.4 in Fig. 3.3-right. The left and right graphs correspond to the unstable and stable modes, respectively. The kinetic energy associated with the unstable perturbation concentrates in the gaseous rivulets, while the opposite occurs in the stable situation. The size of the bubbles resulting from the rivulet breakup is essentially determined by the wavenumber kmax corresponding to the maximum value Ωmax iof the growth factor (Fig. 3.5). In all the cases considered, kmax was at most of order unity, which means that the bubble radius was at least on the order of the strip width R. The quantity kmax (and therefore the bubble radius) was a monotonic function of neither the Weber nor the Reynolds number. In what follows, we will pay attention
3.1. Introduction 45 Figure 3.5: Wave number kmax corresponding to the maximum value of the growth factor, Ωmax i, for an air rivulet surrounded by ethanol with θs0= 120◦. to the dominant mode, and plot the maximum value Ωmax iof the growth factor for unstable systems. Figure 3.6 shows the values of Ωmax ifor an air rivulet surrounded by ethanol. The rivulet is unconditionally unstable for θs0= 120◦. For θs0= 80◦and 89◦, there is a critical Reynolds number below which the rivulet becomes unstable. This instability transition is analyzed in Fig. 3.7, which shows the growth factor Ωias a function of the wavenumber just above and below the critical Reynolds number. Figure 3.6 also indicates that the gaseous rivulet does not stabilize as the Reynolds number decreases, which means that this fluid configuration is unstable even for θs0<90◦ and in the Stokes limit. The above results can also be interpreted in the following way. As explained in Sec. 3.1.2, the Reynolds number does not affect the (dimensionless) zeroth-order solution. Thus, if the rest of the governing parameters remain constant, then so does the basic flow. One can vary the Reynolds number while keeping the rest of the governing parameters constant by changing just the two viscosities. Therefore, the same
46 3. Stability of a rivulet in a microchannel Figure 3.6: Maximum value of the growth factor, Ωmax i, for an air rivulet surrounded by ethanol. basic flow becomes unstable when the gas rivulet (and the coflowing stream) viscosity exceeds a certain critical value. The conclusion is somewhat counterintuitive: despite its dissipative character, viscosity can destabilize the fluid configuration. The fact that the growth rate (damping rate) can increase (decrease) as viscosity increases (while both the basic flow and the rest of parameters remain constant) resembles to a certain extent what happens in other systems. In axisymmetric liquid bridges, the damping rate that characterizes the dominant mode appearing after the eigenfrequency bifurcation also decreases as viscosity increases [87, 88]. In Blasius’ velocity profile over a flat plate, the viscosity effects on the perturbations are neces-
3.1. Introduction 47 Figure 3.7: Growth factor Ωifor an air rivulet surrounded by ethanol. sary to explain the existence of growing modes in a convex velocity distribution [89]. As also occurs in our problem, the maximum value of the growth factor exhibits a non-monotonic dependence with respect to the viscosity for a given base flow. In fact, there is a minimum value of the Reynolds number below which the base flow recovers the stability that characterizes the inviscid case. In all these cases, viscosity appears to produce an extraction of energy from the basic motion in favor of the disturbance. Figure 3.8 shows the maximum value of the growth factor for an ethanol rivulet surrounded by air. The results for θs0= 80◦are not presented because they corre-
48 3. Stability of a rivulet in a microchannel Figure 3.8: Maximum value of the growth factor, Ωmax i, for a ethanol rivulet surrounded by air. spond to stable configurations. Interestingly, the stability condition for θs0= 89◦ and 120◦is the opposite to that of gaseous rivulets: there is a critical Reynolds number above which the rivulet becomes unstable. The system stabilizes for sufficiently low values of the Reynolds number, which is compatible with the predictions obtained from the lubrication theory for liquid rivulets [79, 77, 78]. This conclusion does not apply to the case θs0= 89◦and We = 0.04, where the rivulet was unstable for all the Reynolds numbers considered. 3.2 Conclusions To summarize, we have shown in this chapter the rich topology of the stability map of a rivulet coflowing with a current in a quadrangular channel. The results are qualitatively different from those of similar capillary systems, like cylindrical jets. The contact-line-anchorage condition influences fundamentally the rivulet’s stability. The most noticeable results were obtained for gaseous rivulets. In spite of the dissipative character of viscosity, a basic flow can become unstable when
3.2. Conclusions 49 that quantity exceeds a certain critical value, while the rest of governing parameters remain constant. In fact, gaseous rivulets can be unstable even for θs0<90◦in the Stokes limit. Contrarily to what occurs in the static case, the stability of a flowing liquid rivulet is not determined by its contact angle θs0exclusively. The Reynolds number plays a critical role too. Thus, unstable liquid rivulets with θs0<90◦can be found for large enough Reynolds numbers. A novel microfluidic technique has been recently proposed to produce microbubbles from a gaseous rivulet injected through a T-junction into a microchannel transporting a liquid current [10]. The rivulet breaks up downstream due to a capillary pearling instability, which leads to a quasi-monodisperse collection of microbubbles with diameters on the order of the width of the strip over which the rivulet formed. Decreasing the bubble diameter well below that size would imply the production of unstable rivulets with contact angles well below 90◦. This is an unrealistic task because that occurs within a small parameter region.
Chapter 4 Dissolution of Micro-bubbles 4.1 Introduction The transport in either closed channels[90] or in an open environment present radical differences not only in the obvious mechanical aspects, but most importantly –in the context of our work– in the mass exchange processes between the bubble and the environment. The dissolution of a gas bubble containing a single or multicomponent mixture has been investigated by Chain-Nan Yung et al [91]. They assumed that the gasses inside the bubble are uniform and ideal. Therefore, it will not be necessary to solve coupled equations of Navier-Stokes inside and outside the bubble. They integrated the continuity equation inside the bubble to get the velocity of the interface. Chain solved Navier-Stokes equations and the convection-diffusion equation by using finite difference procedure. Forward difference procedure was used for the time march, while central difference method was used for the space discretization. Up-wind difference was used for the convective term to overcome the problems of nonlinearity. Fumio Takemura et al [15] studied the gas dissolution of a 50
4.2. Problem formulation 57 where Cis a constant, zthe vertical coordinate, Fr =U2 c/gRothe Froude number and the symbol ˙ denotes time derivative. The last term in (4.15) is due to the use of a non-inertial frame of reference. We impose the equilibrium of tangential stress and the kinematic condition, rw rr+uθ r= 0,(4.16) wr−w r+uθ r= 0,(4.17) where uθ= 0 and the velocity in r-direction is constant with respect to θ As a consequences, the boundary condition at the bubble surface, r=R(t), u=˙ R, and wr−w/r = 0.(4.18) Additionally, it has to be considered the convection-diffusion equation to get c(r, θ, t) which is the volume fraction of dissolved gas around the bubble, The differential form of the convection-diffusion equation, ct+v·∇c=D∇2c, (4.19) being each term, v·∇c=ucr+wcθ r,(4.20) ∇2c=2cr r+crr +cθcot θ r2+cθθ r2,(4.21)
58 4. Dissolution of Micro-bubbles So that, ct+ucr+wcθ r=Dcrr +2cr r+cθθ r2+cot θcθ r2,(4.22) The dimensionless convection-difussion equation for the volume concentration in dimensionless form has is, Φt+uΦr+wΦθ r=1 Pe Φrr +2Φr r+Φθθ r2+cot θΦθ r2,(4.23) where the dimensionless concentration of dissolved gas, Φ, is given by, Φ = c−c∞ cso −c∞ ,(4.24) being c(r, θ, t) the mass fraction of gas inside the liquid, cso =cs(0) the initial mass concentration at the surface of the bubble, c∞the initial volume concentration of gas dissolved in the liquid and Pe =RoUc/D the Peclet number, where Dis the gas-liquid molecular diffusivity. The Henry’s law yields the partial pressure of the gas in the liquid p(r, θ, t) as a function of the concentration of gas c(r, θ, t), pp=KHc, where KHis the Henry’s constant. Depending on the definition of the concentration used (i.e.mass concentration, molar concentration, mass fraction, etc.), the physical dimensions of the proportionality constant KHwould vary accordingly. In the present work KH has dimensions of pressure since the concentration is defined as the mass fraction. The mass fraction of gas in the liquid at the bubble surface, cs(t), can be related to the gas pressure inside the bubble as cs(t) = pg(t)/KH, and, in the liquid bulk, it can be written χpa=KHc∞. Therefore, the surface condition for Φ can be written
4.2. Problem formulation 59 as, Φs=cs−c∞ cso −c∞ =pg(t)−χpa pg(0) −χpa =α1+α2H(t),(4.25) where α1=pa−χpa pref and α2=ρgRo pref ,(4.26) being pref =pg(0) −χpa. Away from the bubble, r→ ∞, the velocity matches the external field and the volume fraction the bulk volume fraction, lim r→∞ u=V(t) cos θ, lim r→∞ w=−V(t) sin θand lim r→∞ Φ = 0.(4.27) At any time,V(t), is computed by the integration of the equation of motion for the spherical bubble, which neglecting the mass of the gas inside the bubble writes as follow, 4 3πR3ρg˙ V=−4 3πR3ρgg−Dh−Drag, (4.28) where ρgis the density of the gas inside the bubble, Dhis the static part of the drag and Drag it’s dynamic part, Drag = 2πR2Zπ 0 (−pdRe + 2ur) cos θsin θdθ . (4.29) The static part is, Dh=ZΣ ph~ndσ =ZΩ∇phdΩ,(4.30)
60 4. Dissolution of Micro-bubbles Dh=ZΩ−ρg +ρ˙ VdΩ = 4 3πR3−ρg +ρ˙ V,(4.31) So, 4 3πR3ρg˙ V=4 3πR3ρg −ρgg−ρ˙ V−Drag, (4.32) 4 3πR3(ρ+ρg)˙ V=4 3πR3(ρ−ρg)g−Drag, (4.33) ρgis neglected since ρgρ, resulting, 4 3πR3ρ˙ V=4 3πR3ρg −Drag, (4.34) Finally, the above equation in dimensionless form is, ˙ V=1 Fr 1−Drag 4πR3.(4.35) By applying the mass balance relationship at the bubble-liquid interface[15], d dt 4 3πR3ρg= 2πR2ρlDZπ 0 crsin θdθ, (4.36) The dimensionless form of the previous equation is 2 3 Ro to<T3˙ R(Φspref +p∞) + R˙ Φspref =cso −c∞ Ro ρlDZπ 0 Φr(r=R, θ) sin θdθ, (4.37)
4.2. Problem formulation 61 Or, 2 3pref RoKHUc ρlD<T3˙ R(Φspref +p∞) + R˙ Φspref =Zπ 0 Φr(r=R, θ) sin θdθ, (4.38) finally, the above equation writes in dimensionless form as, 2 3λ[3 ˙ R(Φs+α3) + R˙ Φs] = −Sh,(4.39) where λ=RoBUc/D , being B=KH/(ρl<T) and α3=pp∞ pref . At each time the Smith’ number Shis computed numerically as Sh=−Zπ 0 Φr(r=R, θ) sin θdθ. (4.40) The derivative of the pressure with respect to time ˙ Φs=α2˙ Hresults in, 3˙ R(Φs+α3) + α2R˙ H=−3Sh 2λ,(4.41) Taking into account equation (4.25) and the equation for the evolution of H, ˙ H=−V, (4.42) it is obtained the equation for the time evolution of the radius of the bubble ˙ R=−3/2λ−1Sh+α2RV 3(Φs+α3).(4.43) Present problem must to be closed by the appropriate value of the partial pressure of O2at the upper water-air interface in the tube or, in other words, by the proper
62 4. Dissolution of Micro-bubbles value of the molar fraction of oxygen at the upper interface, χ. If the surrounding atmosphere is renovated with pure air,the value of χshould be close to 0.21 (i.e., pp∞= 0.21pa). However, if the atmosphere is locally enriched by the liberation of oxygen and the renovation is hampered, this value can easily raise, reaching nearly pure oxygen atmosphere concentrations. Thus, when air renovation is not dominated by horizontal convection or mixing, which is the case in the tube over the water-air interface, the proper value would come by imposing a global mass balance between the injected oxygen in the bath and that released by diffusion and convection in vertical direction over the upper free surface. 4.2.2 Numerical procedure Since the size of the bubble is changing with time, the following mapping was made from the physical plane (r,θ,t) to the computational plane (η,ζ,τ), η=r R(t), ζ =θ, τ =t. (4.44) Due to the large differences between the Re and Pe numbers present in the experiments (Re ∼50 and Pe ∼30000) two different computational domains will be used to solve the velocity and pressure fields and the gas concentration in the liquid at the corresponding boundary layers of characteristic width δcand δv, respectively (see figure 4.1.b). For the mechanical problem it is sufficient to truncate the radial domain to an external radius, RV(t) = 30R(t) (ηv= 30), while for solving the concentration boundary layer the domain is truncated at a much closer external radius, Rc(t) = 200Pe−0.5R(t) (ηc= 200Pe−0.5). The time procedure is described next. The time domain is discretized using a
4.2. Problem formulation 63 fixed time step dτ. At given time, τN=Ndτ, the terminal velocity, VN, the height of the bubble, HNand the radius of the bubble, RNare obtained from previous times, τN−1and τN−2, by discretizing in time equations (4.35), (4.42) and (4.43) at time τN−1, using 2nd order central finite differences VN=VN−2+ 2dτ 1 Fr 1−DragN−1 4π(RN−1)3,(4.45) HN=HN−2−2dτV N−1,(4.46) RN=RN−2+ 2dτ −3/2λ−1ShN−1−α2RV N−1 3(ΦN−1 s+α4)−α3/RN−1.(4.47) By the other hand, for the computation of the time evolution of Navier-Stokes equations (4.10)-(4.12) and the convection-diffusion equation (4.23), a mixed implicitexplicit second order projection scheme based on backwards differentiation is employed.[92] Spatial discretization in the (η, ζ) semiplanes employs nηChebyshev spectral collocation points in η, and nζpoints in the ζdirection. These schemes lead to the following set of Helmholtz-type equations: 3Re 2dτ −1 R2 ∂2 ∂η2−2 ηR2 ∂ ∂η +2 η2R2˜uN−1 η2R2∂2 ∂ζ2+cotθ ∂ ∂ζ ˜uN=fu(4.48) 3Re 2dτ −1 R2 ∂2 ∂η2−2 ηR2 ∂ ∂η˜wN−1 η2R2∂2 ∂ζ2+cotζ ∂ ∂ζ +1 sin2ζ˜wN=fw(4.49) 3Pe 2dτ −1 R2 ∂2 ∂η2−2 ηR2 ∂ ∂η˜cN−1 η2R2∂2 ∂ζ2+cotζ ∂ ∂ζ ˜cN=fc(4.50)
64 4. Dissolution of Micro-bubbles where, fu =Re −2convuN−1+convuN−2−1 R ∂p ∂η +4uN−1−uN−2 2dτ ,(4.51) fw =Re −2convwN−1+convwN−2−1 ηR ∂p ∂ζ +4wN−1−wN−2 2dτ ,(4.52) fc =Pe −2convcN−1+convcN−2+4cN−1−cN−2 2dτ ,(4.53) and, convu =1 Ru−η˙ R∂u ∂η +w ηR ∂u ∂ζ −w+2 η2R2Re ∂w ∂ζ +wcotζ(4.54) convw =1 Ru−η˙ R∂w ∂η +w ηR ∂w ∂ζ +u−2 η2R2Re ∂u ∂ζ .(4.55) convc =1 Ru−η˙ R∂c ∂η +w ηR ∂c ∂ζ .(4.56) The previous equations is used to get the predicted values of velocity components and concentration, where the symbol ˜ denotes predicted value, while to get the pressure corrections, the poisson equation is discretized as follows: 1 R2 ∂2 ∂η2+2 ηR2 ∂ ∂η˜pN+1 η2R2∂2 ∂ζ2+cotζ ∂ ∂ζ ˜pN= 1 2dτ 1 ηR ∂˜wN ∂ζ +1 R ∂˜uN ∂η +2 ηR ˜uN+˜wN ηR cotζ. (4.57) This approach allows to use the matrix diagonalization method [93], whose computational cost is of order nη×nζ×min(nη, nζ), to solve the four Helmholtz-type equations (4.48), (4.49), (4.50) and (4.57) resulting from the momentum and concentration equations and the Poisson equation needed to calculate the pressure cor-
4.2. Problem formulation 65 Figure 4.2: Flow chart of the numerical scheme. rections. The nonlinear terms are evaluated using a pseudospectral method.[94] At any time, the required velocity field in the convective term of the discretized concentration equation, is obtained by interpolating the velocity field from the mechanical domain to the concentration domain using a second order interpolation operator along the ηcoordinate. After getting the predicted values, the real value of (ut+1, wt+1, ct+1, pt+1) can be obtained by solving the additional following equations:
66 4. Dissolution of Micro-bubbles pN= ˜pN+pN−1.(4.58) uN= ˜uN−2dτ R ∂˜pN ∂η .(4.59) wN= ˜wN−2dτ ηR ∂˜pN ∂ζ .(4.60) cN= ˜cN.(4.61) A flow chart of the numerical scheme used in the present work is depicted in Fig.4.2. Note that both the spectral resolution used and the two different computational domains considered, allow us to use a much less number of grid points that using standards finite-differences in a single mesh. Therefore, we have carried out the numerical simulations in a grid with nη= 60 and nζ= 25 for the cases presented in this study. The time step employed in the simulations was ∆τ= 0.005, since no significant differences in the temporal evolution of the flow were found by using smaller time steps. 4.3 Experimental setup 4.3.1 Apparatus Configuration It has been prepared an experimental setup in order to validate the numerical code. The experimental setup is sketched in figure 4.3. Bubbles of oxygen, liberated at the bottom of a test section filled with quiescent distilled water, ascends freely through the test section. The test section consists in a tube of 9 mm diameter and 116 cm
4.4. Results and conclusions 73 of the process before the bubble reaches the quasi-terminal velocity. For example, figure (4.7) show contours of the dynamic pressure of the liquid around the bubble at different times. It can be seen how during this short period of time the change in the bubble radius is negligible. On the other hand, the contour of the concentration depicted in figure (4.8) show the development of a quite thin concentration boundary layer around the sphere with its corresponding wake. The correct computation of this boundary layer is critical to get an accurate value of the Smith’ number Sh (equation 4.39) and, consequently, to get a good estimation of the change of the bubble radius with time (equation 4.43). In our experimental setup the positions at which the radius of the bubble are measured are fixed. Therefore, in other to compare the experimental results and the numerical ones, it is better to plot the radius of the bubble as a function of the bubble’s deep instead of plotting the radius versus the time. Note that from a numerical point of view, the time and the bubble’s deep are relate through equation (4.42). The results for both case are shown in in figure (4.9). The solid lines correspond to the numerical prediction while the solid (open) symbols correspond to the experimental mean (standard deviation) values of the 20 series of experimental measurements. To conclude, a tracking-interface numerical method was used to achieve this work. The method has been applied to study the isothermal dissolution of small single rising bubble. The key elements of the method are the use of a frame of reference moving with the bubble and the application of different meshes to solve the mechanical and mass-diffusion problems. An experiment of a small oxygen bubble rising in water has been carried out with remarkable agreement to the numerical model.
74 4. Dissolution of Micro-bubbles 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.02 0.04 0.06 0.08 0.1 t(s) V(m/s) 0 1 2 3 4 5 6 7 8 0 0.02 0.04 0.06 0.08 0.1 t(s) V(m/s) (b) (a) Figure 4.6: Time evolution of the velocity of the bubble for experimental series I: a) Short time evolution (time scale to reach a quasi-terminal velocity tT); b) Long time evolution (time scale tR)
4.4. Results and conclusions 75 (c) (a) (b) (d) Figure 4.7: Pressure distribution around the bubble for case I at different times; a) t=0.0122 s, b) t=0.0245 s, c) t=0.0612 s and d) t=0.1224 s. (a) (b) (c) (d) Figure 4.8: Concentration contours around the bubble for case I at different times; a) t=0.0122 s, b) t=0.0245 s, c) t=0.0612 s and d) t=0.1224 s.
76 4. Dissolution of Micro-bubbles 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8x 10-4 Deep (m) Radius (m) case II case I Figure 4.9: Radius of the bubble as a function of the deep for the two experiments, the solid lines correspond to the simulations results while the closed (open) circles correspond with the mean (standard deviation) experimental values.
Chapter 5 Electrical disruption of pendant liquid drops 5.1 Introduction The study of steady cone-jets of liquids in air re-emerged vigorously in the decade of the 90 as a result of the important findings of Fenn et al[97]. In the wake of this impulse, Barrero et al. [98] showed experimentally that, the electrodispersion of conducting liquids into non-conducting liquids in the steady cone-jet mode is possible. The cone-jet, understood as a technique to produce a continuous stream of monodisperse micro(nano) droplet, broaden ,therefore, its use to new fields of interest[99]. For example, liquid-liquid cone-jet electrospraying, in accordance to low surface tension values in liquid-liquid interfaces, requires lower states of electrification to be formed being, therefore, less hazardous the handling of biological or biomedical agents[100]. 77
78 5. Electrical disruption of pendant liquid drops Recent advances in analytical chemistry pursue minimal sample consumption by restricting the analysis to just the first droplets (or droplets) selectively withdrawn by electrical means[101]. The same selective and precise “on-demand” manipulation by electrical means is also an objective in microfluidic liquid-liquid devices[102]. It is worth, therefore, to gain a deeper insight on the physic of formation of this primary electrically generated droplet. Previous works has been conducted to study related issues. The tip streaming arising in low-conductivity drops after a step change in the electric field magnitude in a dynamically neutral atmosphere has been examined both numerically[103] and experimentally [104, 105, 106, 107, 103]. The size and charge of this primary droplet generated in air has been also object of interest. Numerical[108, 109] and experimental[109] studies have been used to derive scaling laws for this primary electrified droplet. In the present chapter we aim to study both numerically and experimentally the dynamic of suddenly electrified droplets focusing in the dynamical role played by an non-negligible outer incompressible atmosphere. It is well known that the applied electric field plays a major role in the dynamics[110]. Not surprisingly, after the sudden onset of a subcritical electric field the more elongated equilibrium position is reached after the oscillations have died out. The supercritical dynamic is more rich and varied. Beside to the well-known tip streaming mode appears others modes shown in figure 5.3 like the splitting mode and the splashing mode. The transition from the splitting mode to the tip streaming in the case of a conducting droplet of lower viscosity than the outer dielectric medium was analyzed in previous works[111, 112]. In the present chapter we will focus in characterizing the splashing mode.
5.2. Formulation of the problem 79 V H ri ,mi ,ki ,ei ro ,mo ,ko=0,eo R Rh r x Z(x) Figure 5.1: Sketch of the problem. The red box shows the computational domain. 5.2 Formulation of the problem The geometrical and electrical configuration considered in this work is sketched in Fig. 5.1. A certain liquid droplet of volume Ω pends of a infinity plate being the rim of the droplet anchored at a distant Rof the axis of symmetry. A second infinity plate is faced in parallel at the distance H. The gap between these parallel plates is filled with a second fluid immiscible with the droplet. The droplet fluid properties are the density, ρi, viscosity µi, electrical conductivity κiand permittivity εiwhile the corresponding properties for the surrounding phase are formed by substituting the subscript “i” by “o”. Both fluids are assumed to be incompressible but the surrounding fluid is not conducting (κo= 0). The surface tension between fluids is denoted by γ. At a certain instant a drop of voltage, V, (or, equivalently, an electric field Eo=V/H) is onset between the plates. The dynamical behaviour of the droplet
80 5. Electrical disruption of pendant liquid drops after the sudden electrification is determined by the magnitude of Eo. 5.2.1 Numerical model 5.2.1.1 Governing equations Equations of the model below are written in dimensionless form using as characteristic quantities the outer permittivity εo, the rim radius R, the surface tension γ and the inner density, ρi. We use the EHD-Volume of Fluid (VoF) solver developed by Lopez-Herrera et al.[113] for the GERRIS platform[114]. This solver has been thoroughly tested in previous works[115, 103, 109]. Since VoF method treats immiscible fluids as a single one with spatially varying properties through the interface, the governing equations for both fluids writes, ∇·v= 0 ,(5.1) ˜ρv t+v·∇v=−∇p+∇·˜µ∇v+∇vT+∇·˜εEE −E·E 2I+Bo x+δsζn (5.2) and ∇·(˜ε∇ϕ) = −qwith E=−∇ϕ , (5.3) where v(x, t), p(x, t) and ϕ(x, t) are the velocity field, pressure distribution and electric potential in the computational domain, respectively. Eis naturally the electric field. The gravitational forces, orientated in direction x, results, after nonadimensionalization, in the Bond number Bo =ρigR2/γ. The third term in the r.h.s of Eq. (5.2) corresponds to the electric stresses and results of applying the divergence of the Maxwell stress tensor where Idenotes the unity tensor.
5.2. Formulation of the problem 81 The last term in the momentum equation (5.2) corresponds to the surface tension term that acts only at the interface (δsis the Dirac delta). ndenotes the unitary vector normal to the interface being ζthe interface curvature. q(x, t) is the dimensionless volume charge density that obeys to the conservation equation, q t+∇·(qv) = ∇·(−˜κE).(5.4) Observe that qhas only sense in the inner conducting fluid phase being strictly zero in the outer fluid phase given its dielectric (non-conducting) nature. ˜ρ(x, t), ˜µ(x, t), ˜ε(x, t) and ˜κ(x, t) stand for the spatial varying dimensionless fluid properties, density, viscosity, electrical permittivity and conductivity, respectively, ˜ρ= Ψ + ρr(1 −Ψ) ,(5.5) ˜µ=Cµ(Ψ + µr(1 −Ψ)) ,(5.6) 1 ˜ε=1 βΨ+1 1−Ψ,(5.7) and ˜κ=α β Ψ.(5.8) being ρrand µrthe outer to inner ratio of densities and viscosities, respectively, ρr=ρo/ρiand µr=µo/µi.Cµis the Ohnesorge number based on the inner fluid properties, Cµ=µi/(ρiRγ)1/2.βis the relative permittivity, β=εi/εo.αis the relaxation parameter[116], α= (κ2R3ρi/γε2 i)1/2. Note that αcoincides with the ratio of the capillary time tc= (R3ρi/γ)1/2to the electrical relaxation time te=εi/κi. Finally, Ψ(x, t) is the volume fraction that serves to track the interface position.
82 5. Electrical disruption of pendant liquid drops Ψ(x, t) is governed by the equation, Ψ t+∇·(Ψv)=0.(5.9) 5.2.1.2 Computational domain and boundary conditions Profiting of the axisymmetric character of the problem, the computational domain has been restricted to the red rectangle (H/R)×(R∞/R) shown in Fig. 5.1. Trying to reproduce as much as possible the experimental setup described below we have set H/R to 4.29 and R∞/R has been fixed to 8.6. We have checked that the enlargement of the computational domain (R∞>8.6R) does not have sense since the dynamic of the droplet is unaffected. The degree of the electrification level is measured by the dimensionless electric field, χgiven by, χ=RεoE2 o γ1/2 ,(5.10) Therefore, this electrification level is obtained by imposing a constant dimensionless voltage to the upper plate equal to ϕ(0, r;t) = Hχ/R while the downstream plate is kept grounded. Far away of the droplet, at the boundary r=R∞/R, the voltage drop is assumed to be linear and the fluid velocity is considered negligible. Known the density of the fluids, ρoand ρi, and the volume of the inner fluid Ω, it is calculated the dimensionless position of the interface, r=Z(x), imposing equilibrium between gravitational/buoyacy forces and surface tension forces and the
5.3. Results and discussion 89 5.3.1 Subcritical regime t 0 5 10 15 20 Zapex 1.15 1.2 1.25 1.3 1.35 1.4 1.45 Figure 5.4: Temporal evolution of the dimensionless apex position Z(t) for ϑ= 1.26, Cµ= 4.1×10−3,Borel = 2.254, χ= 1.46, ρr= 0.912 , µr= 4.54. The symbols and solid line are the experimental and the numerical results, respectively. The liquid flow in the equipotential drop is originated by the appearance of electric stresses at the free surface. Both the electric charge density and the electric stress reach their maximum values at the apex. The electric stress is balanced by the hydrostatic and capillary pressures (viscous stresses normal to the free surface can be generally neglected). Therefore, the reduced pressure decreases at the apex,
90 5. Electrical disruption of pendant liquid drops and the liquid is suctioned towards that region. This liquid motion stretches the drop, increasing the free surface curvature at the apex. Consequently, a restoring capillary force appears in that region. If the electric stresses are not high enough to overcome the surface tension force, then the drop reaches a new equilibrium state. In principle, the new equilibrium shape could be reached through either an overdamped extensional deformation or the damping of free surface oscillations. Figure 5.4 shows the temporal evolution of the apex position,Zapex, of a subcritical drop.
5.3. Results and discussion 91 5.3.2 Supercritical regime: The splashing mode 0.5mm 0.5mm 0.5mm 0 c tt ( st 0 ) 05.1 c tt ( st 017.0 ) 15.1 c tt ( st 0186.0 ) Figure 5.5: The droplet splashing at different stages. The right column shows the numerical predictions. Green lines correspond to electric isopotential lines. Dimensionless values for this case are: ϑ= 0.64, Cµ= 4.1×10−3,χ= 0.39, ρr= 0.908 and µr= 4.04.
92 5. Electrical disruption of pendant liquid drops Different instants of a droplet breaking in the splashing mode are shown in figure 5.5. The splashing mode is characterized by two stages: (i) the droplet is slowly stretched in axial direction up to a certain neck radius, dn(and length, Ls) is reached (ii) the front of the droplet, then, expands faster in radial direction that in axial direction forming a jellyfish splashing structure. The characteristic time for both stage is similar. Interestingly, in some cases this “jellyfish” splashing structure is repeated in cascade (see figure 5.3, C). At first glance, the early stage of axial stretching is common in all modes. This first stage is essentially the result of the competition between the electric pressure and surface tension being the drop regarded as equipotential. However, the second stage is radically different in tip-streaming mode and splashing mode from the electrical point of view. While the jetting in the tip streaming can be seen as a consequence of the finite conductivity of the fluid[121, 103], the splashing occurs in times of the order of the capillary time behaving the fluid as equipotential since electric relaxation time is much shorter. Consequently, the conductivity and permittivity of the inner fluid are unimportant in the splashing mode. We have assessed the conducting character of the fluid by means of the numerical computations (see electric isopotential lines in figure 5.5).
5.3. Results and discussion 93 )( A )( B )( C 1mm A B C Figure 5.6: Effect of droplet volume on splashing size: (A) Ω = 17.77mm3(B) Ω = 21.28mm3(C) Ω = 25.35mm3.
94 5. Electrical disruption of pendant liquid drops 0.01 0.1 1 10 611 16 21 26 31 dn(mm) Ω(mm3) v=2100 volts v=2300 volts v=2500 volts Figure 5.7: Neck diameter Vs. droplet volume for voltage 2100,2300 and 2500. Some other differences can be pointed out between the modes tip-streaming and splashing. Tip streaming is of local (and universal) nature. The ejected jet obeys to universal scaling laws based on the liquid properties[108, 103, 109] being independent of large scale variables as the applied voltage, the droplet radius or the particular geometrical configuration used (pendant from a plate, levitating or pending from a needle, for example). On the contrary we found that either the applied voltage and the droplet volume affect the form of the splashing. This effect can be seen clearly in figure 5.6. Also, figure 5.7 shows the change of the neck diameter dnwith the droplet volume Ω and the applied voltage V. These large scale variables have the same effect on other modes (Split-Splashing and Splitting).
5.3. Results and discussion 95 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0,1 1 10 dn/Ro Ls/Ro µr= 0.78 µr= 4.04 µr= 14.25 µr= 71.96 0.8 1 1.2 1.4 1.6 1.8 2 10−1 100 101 ϑ Ls/Ro µr= 0.78 µr= 4.04 µr= 14.25 µr= 71.96 AB Figure 5.8: Non dimensional splash length, Ls/R, versus: (A) The dimensionless volume, ϑ, (B) the dimensionless neck diameter, dn/R. Experiments shown cover two orders of magnitude of the viscosity ratio, µr∈[0.78,71.96]. Moreover some interesting remarks arise from experimental data. In figure 5.8 we plot the dimensionless splash length, Ls/R, as a function of the dimensionless volume ϑ(subplot A). In subplot B it is explored the relationship between the splash length and the neck radius. A first aspect worth to be noted is the negligible influence of the outer viscosity in the splashing process. The reason for this negligible effect of the viscosity ratio can be attributed to the relatively large characteristic hydrodynamic time of the process. Despite the high values of the outer viscosity the splashing developing is slow enough to have unimportant outer viscosity forces compared to electric or capillary ones. The effect of the outer medium is, some how, paradoxical, the splashing mode develops because the outer medium is not negligible from the dynamic point of view, but experiments shows that the outer viscosity is trivial in the formation of the splashing. On the contrary, it would expect that the density ratio ρrwere significant in the process. However, it is difficult to verify this point
96 5. Electrical disruption of pendant liquid drops experimentally since the density ratio can not be varied too much with the present experimental setup. It seems also to exist a decoupling between the splashing length and the neck radius (see figure 5.8, subplot B). Summarizing, experiments suggest that the splash length is an only function of the dimensionless volume ϑ. The dimensionless neck radius has to be, necessarily, a function of all the dimensionless parameters governing the problem, dn R=f(ϑ, χ, Cµ, ρr, µr, β, α, H/R),i.e. (5.12) the dimensionless volume, electrification level, inner viscosity; relative density, viscosity and permittivity; electrical conductivity and distance between plates, respectively. Since we claim (supported by the numerical simulations) that the droplet can be assumed equipotential along the first stages of the splashing, either the relative electrical permittivity βand dimensionless conductivity, α, are irrelevant. Also, since the experimental setup is fixed and the ratio of densities in our experiments are very close to the unity either the width of the gap H/R and the relative density can be also safely dismissed in relation (5.12). Finally, we assume that the relative viscosity µr, as in the case of the splash length, is negligible. Thus, the relation (5.12) reduces to, dn R≃f(ϑ, χ, Cµ) = f(ϑ1/3χ2/3C−1/5 µ).(5.13) In the above functional relationship (5.13) we have assumed that the dimensionless neck radius scales with the single parameter, ϑ1/3χ2/3C−1/5 µbeing the exponent those that better suited the experiments. In figure 5.9 we plot the dimensionless neck radius as a function of the scaling proposed in relation (5.13) using the same data
5.4. Conclusions 97 of figure 5.8. 2 2.2 2.4 2.6 2.8 3 0 0.2 0.4 0.6 0.8 ϑ1/3 χ2/3/Cµ 1/5 dn/R Figure 5.9: Non dimensional splash neck radius, dn/R, versus the scaling ϑ1/3χ2/3C−1/5 µ. 5.4 Conclusions To conclude, intensive experiments have been carried out in order to study all parameters that could affect on the electrical disruption of pendant conducting liquid drop in a dielectric liquid. Various conducting and dielectric liquids have been used to study the inner and outer, liquid medium properties, effect on our drop dispersion. Experiments have showed two regime; subcritical regime, when the drop oscillate to a stable equilibrium position just like in the case of surrounding air atmosphere. The second one is a supercritical regime, when the electrical force overcome cap-
98 5. Electrical disruption of pendant liquid drops illary forces and a very fast microjet is emitted. The last behaviour is known as tip streaming mode. But, our results show that this is not the only mode in the super-critical regime. We have found out that other modes are exist. With changing the outer medium viscosity three more modes have been revealed. With low outer medium viscosity appears tip streaming mode. Splashing mode is shown when the outer viscosity is moderate (5 and 20 cSt). Split-Splashing is observed with high viscosity (100 cSt). While, with very high viscosity (10000 cSt) appears the splitting mode. Numerical model has been made to simulate the behavior of the electrified pendant drop. A great agreement was observed between the experimental and numerical results. Depending on massive experimental results, a scaling law has been obtained to characterize the neck diameter in the splashing mode. It was found that the neck diameter in the split-splashing mode also obey the scaling law. This study has a great importance. It is not only helping to get a better understanding of an interesting topic like liquid-liquid dispersion, but also it estimates the threshold limitation of producing micro-droplets in another liquid considering the conducting and the dielectric liquid properties. Estimating such limitation has a great impact on application like micro-encapsulation.
6.3. Numerical simulations 105 v p R 0 D /2 H Figure 6.6: Simulation domain and numerical mesh mesh size. The interface between the two phases was tracked by solving a continuity equation for the volume fraction of one of the phases. This calculation was performed by using an explicit time-marching scheme, while the rest of the equations were solved implicitly. The time step ∆twas sufficiently small to ensure that the global Courant number Co =vm∆t/∆ybased on the mean velocity vmin the cell and the cell size ∆ywas much less than unity. Regarding the spatial discretization of the equations, the third-order modified MUSCL scheme [131] was used to obtain the face fluxes whenever a cell was completely immersed in a single phase. When the cell was near the interface, the GEO-RECONSTRUCTION algorithm was used. The pressure corrections were computed with the bodyforceweighted scheme, and the pressure-velocity coupling in a segregated solver was treated with the PISO method [132]. All the simulations were conducted with
106 6. Small drops from large nozzles D= 30mm,H= 20mm and R0= 1mm. It was noticed from experiments that the size of the drops generated is very sensitive to the interface meniscus. Also, we can see in figure 6.7, numerical simulation of the evolution of the interface for plan meniscus (left) and concave meniscus (right) with a produced droplet of 116µm and 126µm respectively. It is obvious that the meniscus form is an important parameter, therefore, the profile of the meniscus was modified to be as in the experiment. Figure 6.7: Simulation showing the influence of the initial meniscus form. At left (plane meniscus) and at right (concave meniscus).
6.4. Results and discussion 107 6.4 Results and discussion 6.4.1 Experimental jetting process In this study, a mixture of water+ 60% wt glycerol and Silicone oils of 5 and 10 cSt were used as working liquids. The physical properties are listed in table 5.1. The input pulse width were set to be tpw = 3 and 4ms. Table 6.1: Physical properties of working liquids Liquid ρ(kg/m3)µ(cP)γ(mN/m) water+ 60% wt glycerol mix. (5cSt) 1116 5 61.9 Silicone oil(5cSt) 912 4.2 21.9 Silicone oil(10cSt) 936 8.7 23.1 In each experiment, the sine pulse amplitude and width were introduced by LabView program. The input pulse and the measured displacement of the piston are illustrated in figure 6.8(a). While the corresponding velocity and pressure inside the reservoir are shown in figure 6.8(b). When the negative pulse is applied, the free surface of the meniscus is pulled back into the reservoir forming a cavity. Then, the piston moves into the other direction making a positive pressure a round the cavity. The cavity collapses forming a very thin jet smaller than the orifice. Finally, a drop as the same order of the jet separates and the meniscus returns back to the equilibrium. Figure 6.9 shows Experimental observation of the jetting process of silicone oil (10 cSt). Experimental results shows that many parameters affect on the produced drop size. In figure 6.10 the effect of the pulse amplitude can be seen. Increasing the velocity amplitude of the piston makes the cavity collapse faster as the kinetic energy
108 6. Small drops from large nozzles Figure 6.8: (a) The input pulse (dashed line) and the measured displacement of the piston (solid line). (b) The pressure inside the reservoir(solid line) and the velocity of the piston (dashed line). that transfers to the fluid around the nozzle increases which leads to a thinner jet and a smaller droplet. Also, the speed of of the jet (and the drop) increase at faster collapse. The introduced kinetic energy to the system does not affect only on the droplet size and it’s speed, but also it characterizes the mode where only one droplet could be produced. Beside the piston velocity, there are other important parameters. At figure 6.11, we plot the generated drop diameter versus the velocity of the piston
6.4. Results and discussion 109 mst 27.2 ms t4 ms t22.5 mst 8 1 mm Figure 6.9: Jetting process of Silicone oil 10cSt from a 2 mm diameter nozzle for various working liquids in order to study the effect of the velocity amplitude, pulse width and liquid properties on the final droplet size. And also to identify the range of velocity where we can produce only one droplet for each working liquid. It is obvious that the drop size decreases with the increasing of the velocity amplitude. After a certain amplitude, more than one drop are generated. A wider rang of drop size was achieved with lower viscosity. Also, it was observed that a lower surface tension helps to generate smaller drop. It was managed to generate a drop of silicon oil (5cSt) with a radius of 27µm , being the surface tension 21.9mN/m. Also it was noticed that the pulse width tpw does not have a great influence if we have used the positive velocity amplitude vpto be the piston velocity parameter.
110 6. Small drops from large nozzles smmVp/7.2 smmVp/79.2 smmVp/84.2 150 μm Figure 6.10: Images of Drops generated with different piston velocity 6.4.2 Numerical to experimental results In our numerical model, we assumed that the reservoir has only one inlet/exit to the fluid which is the nozzle, while in the experiment we have also the syringe that is connected to the reservoir and exposed to the atmosphere. So, we assumed in the simulation that the volume of the formed cavity (Vcavity) is equal to the volume of liquid moved by the piston (Vpiston), while in the experiment (Vpiston) equal to (Vcavity) and the volume displaced inside the syringe. Also, in the simulation the nozzle is a rigid solid that does not deform and as it was mentioned before, the upper plate of the print-head prototype in the experiment has a thickness of 0.25mm. So, there is a chance that the plate deforms with the movement of the piston. Therefore, to make sure that our model simulate the same condition of the experiment, we had compared (Vpiston) with (Vcavity) of the experiment during the pulse period. Vpiston was calculated by multiply the displacement of the piston by the area of the reservoir. While, Vcavity was measured by detecting the border of the cavity from the images using an image processing program (ImageJ).
6.4. Results and discussion 111 0 50 100 150 200 250 300 350 0 0.001 0.002 0.003 0.004 0.005 d (μm) vp(m/s) g/w 5cSt-4ms silicone oil 10cSt-3ms silicone oil 10cSt-4ms silicone oil 5cSt-4ms Figure 6.11: Drop size versus piston velocity with different governing parameters It is clear from figure 6.12 that there is discrepancy between the volume of the cavity and the volume moved by the piston. So, if we have used in the simulation the same piston velocity that is in the experiment, we will have a higher kinetic energy around the nozzle. So, the droplet size in the simulation will be smaller than the one in the experiment. The cavity volume in the experiment ≃0.67Vpiston. So, depending on this result the piston velocity could be correlated in our simulation to give the same cavity volume as follow: 0.67Vpiston =AZvpdt. (6.1)
112 6. Small drops from large nozzles vp= 0.67dh dt .(6.2) being, Ais the reservoir cross sectional area and his the piston displacement. 0 1 2 3 4 5 6 x 10 -3 -5 -4 -3 -2 -1 0 1x 10 -9 Time (s) Vpiston Vcavity Figure 6.12: Comparison between (Vpiston) and (Vcavity) ms t24.2 ms t7.3 ms t2.7 ms t9.9 Figure 6.13: Numerical simulation of Jetting process for Silicone oil 10cSt at pressure amplitude 232 Pascal and pulse width 4 ms Figure 6.13 shows numerical simulation of the jetting process of silicone oil (10cSt) at pressure amplitude 232 Pascal and pulse width 4 ms. The numerical
6.4. Results and discussion 113 simulation has a good agreement with the experiment. In figure 6.14, we plot experimental and numerical results of the droplet diameter front the velocity amplitude. The results illustrate the efficiency of our numerical model and the reliability of the piston velocity correlation. 0 50 100 150 200 250 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 d (μm) vp (mm/s) experiment-(SO10cSt)-4ms experiment-(SO5cSt)-4ms simulation-(SO10cSt)-4ms simulation-(SO5cSt)-4ms Figure 6.14: Comparison between experimental and numerical results 6.4.3 Scaling analysis In this section, an analysis has been made to obtain scaling law for the drop size generated by (LS) print-head prototype. From the previous results we can say that the dimensionless droplet diameter is a function of the following dimensionless parameters, d R0 =f(vp/vc, oh, tp/tc, H/R0, D/R0),i.e. (6.3)
114 6. Small drops from large nozzles 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5 5.5 6 6.5 7 7.5 8 d/R0 (vp/vc)-1 oh0.5 (tp/tc)2 g/w 5cSt silicone oil 10cSt silicone oil 5cSt Experimental fitting Figure 6.15: Dimensionless diameter of the drop d/R0versus the scaling groups (vp/vc)−1oh0.5(tp/tc)2 Being, capillary velocity vc= (σ/ρR0)1/2, ohnesorge number oh =µ/√ρgR0σand capillary time tc= (ρR3 0/σ)1/2. In relation (6.3), beside positive velocity amplitude, liquid properties and geometrical parameters, we have introduced a new parameter which is (tp) the width of the positive part in the velocity pulse (see figure 6.8(b)). Although we have seen in section (6.4.1) that the pulse width tpw does not have significant effect on the droplet size, we found that tphave a great influence. It was shown from the piston velocity measurement for each experimental point, that tpis directly related to liquid properties as the positive part of the pulse is considered to be the system response to the negative applied part. Our numerical simulation showed that Hand Ddo not have influence on the drop size which make sense as H, D ≫R0. So, H/R0and D/R0can be omitted and the relation (6.3) reduces to, d R0≃f(vp/vc, oh, tp/tc) = f((vp/vc)−1, oh0.5,(tp/tc)2),i.e. (6.4)
LIST OF FIGURES 121 6.2 pull-push and pull-push-pull waveform pulse [129] . . . . . . . . . . . 100 6.3 Configuration of (LS) prototype and it’s jetting process [130] . . . . . 101 6.4 Region in which droplers can be generated [130] . . . . . . . . . . . . 102 6.5 Experimentalsetup............................103 6.6 Simulation domain and numerical mesh . . . . . . . . . . . . . . . . . 105 6.7 Simulation showing the influence of the initial meniscus form. At left (plane meniscus) and at right (concave meniscus). . . . . . . . . . . . 106 6.8 (a) The input pulse (dashed line) and the measured displacement of the piston (solid line). (b) The pressure inside the reservoir(solid line) and the velocity of the piston (dashed line). . . . . . . . . . . . . . . 108 6.9 Jetting process of Silicone oil 10cSt from a 2 mm diameter nozzle . . 109 6.10 Images of Drops generated with different piston velocity . . . . . . . 110 6.11 Drop size versus piston velocity with different governing parameters . 111 6.12 Comparison between (Vpiston) and (Vcavity) ...............112 6.13 Numerical simulation of Jetting process for Silicone oil 10cSt at pressure amplitude 232 Pascal and pulse width 4 ms . . . . . . . . . . . . 112 6.14 Comparison between experimental and numerical results . . . . . . . 113 6.15 Dimensionless diameter of the drop d/R0versus the scaling groups (vp/vc)−1oh0.5(tp/tc)2...........................114
List of Tables 4.1 Experimental conditions . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1 Interfacialtension............................. 85 5.2 Physicalproperties ............................ 86 6.1 Physical properties of working liquids . . . . . . . . . . . . . . . . . . 107 122
Bibliography [1] M. M. Denn. Continuous drawing of liquids to form fibers. Ann. Rev. Fluid Mech., 12:365–387, 1980. [2] Saeid Vafaei and Dongsheng Wen. Bubble formation on a submerged micronozzle. Colloid and Interface Science, 343:291–297, 2010. [3] C. S. Smith. On blowing bubbles for bragg’s dynamic crystal model. J. Appl. Phys., 20:631–632, 1949. [4] A. M. Ga˜n´an-Calvo and J. M. Gordillo. Perfectly monodisperse microbubbling by capillary flow focusing. Phys. Rev. Lett., 87:274501, 2001. [5] A. M. Ga˜n´an-Calvo. Perfectly monodisperse microbubbling by capillary flow focusing: An alternate physical description and universal scaling. Phys. Rev. E, 69:027301, 2004. [6] P. Garstecki, I. Gitlin, W. DiLuzio, G. M. Whitesides, E. Kumacheva, and H. A. Stone. Formation of monodisperse bubbles in a microfluidic flow-focusing device. Appl. Phys. Lett., 85:2649–2651, 2004. [7] T. Thorsen, R.W. Roberts, F. H. Arnold, and S. R. Quake. Dynamic pattern 123
124 BIBLIOGRAPHY formation in a vesicle-generating microfluidic device. Phys. Rev. Lett., 86:4163, 2001. [8] A. Gunther, S. A. Khan, M. Thalmann, F. Trachsel, and K. F. Jensen. Transport and reaction in microscale segmented gas–liquid flow. Lab Chip, 4:278, 2004. [9] M. A. Herrada, A. M. Ga˜n´an-Calvo, and J. M. L´opez-Herrera. Generation of small mono-disperse bubbles in axisymmetric t-junction: The role of swirl. Phys. Fluids, 23:072004, 2011. [10] M. A. Herrada, A. M. Ga˜n´an-Calvo, and J. M. Montanero. Theoretical investigation of a technique to produce microbubbles by a microfluidic t junction. Phys. Rev. E, 88:033027, 2013. [11] D. W. Moore. The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech., 23:749–766, 1965. [12] P. C. Duineveld. Rise velocity and shape of bubbles in pure water at high reynolds number. Journal of Fluid Mechanics, 292:325–332, 1995. [13] G. Mougin and J. Magnaudet. Path instability of a rising bubble. Physical Review Letters, 88:145021–145024, 2002. [14] C. Mart´ınez-Baz´an, J. L. Monta˜n´es, and J. C. Lasheras. Statistical description of the bubble cloud resulting from the injection of air into a turbulent water jet. International Journal of Multiphase Flow, 28:597–615, 2002. [15] Fumio Takemura and Akira Yabe. Gas dissolution process of spherical rising gas bubble. Chemical Engineering Science, 53(15):2691–2699, 1998.
BIBLIOGRAPHY 125 [16] F. Takemura and A. Yabe. Rising speed and dissolution rate of a carbon dioxide bubble in slightly contaminated water. Journal of Fluid Mechanics, 378:319–334, 1999. [17] D.F. McGinnis, J. Greinert, Y. Artemov, S.E. Beaubien, and A. W¨uest. Fate of rising methane bubbles in stratified waters: How much methane reaches the atmosphere? Journal of Geophysical Research C: Oceans, 111:C09007, 2006. [18] H. Ding, P. D. M. Spelt, and C. Shu. Diffuse interface model for incompressible two-phase flows with large density ratios. Journal of Computational Physics, 226:2078–2095, 2007. [19] M.G. Gerritsen and L.J. Durlofsky. Modeling fluid flow in oil reservoirs. Annual Review of Fluid Mechanics, 37:211–238, 2005. [20] A. Shokrollahi, M. Arabloo, F. Gharagheizi, and A.H. Mohammadi. Intelligent model for prediction of co2 - reservoir oil minimum miscibility pressure. Fuel, 112:375–384, 2013. [21] V.L. Singleton and J.C. Little. Designing hypolimnetic aeration and oxygenation systems - a review. Environmental Science and Technology, 40:7512–7520, 2006. [22] D.N. Rao and J.I. Lee. Determination of gas-oil miscibility conditions by interfacial tension measurements. Journal of Colloid and Interface Science, 262:474–482, 2003. [23] A. M. Ga˜n´an-Calvo. Enhanced liquid atomization: From flow-focusing to flow-blurring. Appl. Phys. Lett., 86:214101, 2005.
126 BIBLIOGRAPHY [24] J. Zeleny. Instability of electrified liquid surfaces. Phys. Rev., 10:1–6, 1917. [25] G. Taylor. Disintegration of water drops in electric field. Proc. R. Soc. Lond. A, 280:383–397, 1964. [26] A. Watanabe, K. Higashitsuji, and K. Nishizawa. Studies on electrocapillary emulsification. J. Colloid Interf. Sci., 64:278–289, 1978. [27] M. Sato, M. Saito, and T. Hatori. Emulsification and size control of insulating and/or viscous liquids in liquid-liquid systems by electrostatic dispersion. Journal of Colloid And Interface Science, 156(2):504–507, 1993. [28] C. Tsouris, D. W. DePaoli, J. Q. Feng, 0. A. Basaran, and T. C. Scott. Electrostatic spraying of nonconductive fluids into conductive fluids. AIChE J., 40:1920–1923, 1994. [29] Masayuki Sato, Toshihiro Hatori, and Masahiro Saito. Experimental investigation of droplet formation mechanisms by electrostatic dispersion in a liquid–liquid system. IEEE Transactions on Industrial Applications, 33:1527– 1534, 1997. [30] Costas Tsouris, Won-Tae Shin, and Sotira Yiacoumi. Pumping, spraying, and mixing of fluids by electric fields. The Canadian Journal of Chemical Eng., 76:589–599, 1998. [31] G. Gneist and H.-J. Bart. Electrostatic drop formation in liquid/liquid systems. Chemical Engineering and Technology, 25:899–904, 2002. [32] J. M. Grace and J. C. M. Marijnissen. A review of liquid atomization by electrical means. J. Aerosol Sci., 25(6):1005–1019, 1994.
BIBLIOGRAPHY 127 [33] M. Doring. Ink-jet printing. Philips Tech. Rev., 40:192, 1982. [34] B. Nivi B. A. Ridley and J. M. Jacobson. All-inorganic field effect transistors fabricated by printing. Science, 286:746, 1999. [35] T. P. Theriault K. Konrad E. Lachenmeier M. Schena, R. A. Heller and R. W. Davis. Microarrays: Biotechnology’s discovery platform for functional genomics. Trends Biotechnol, 16:301, 1998. [36] S. I. Zoltan. Pulsed droplet ejection system. U. S. Patent No. 3,683,212, 1972. [37] E. L. Kyser and S. B. Sears. Method and apparatus for recording with writing fluids and drop projection means therefore. U. S. Patent No. 3,946,398, 1976. [38] K. Noboru Kobayashi, H. and S. Ohno. Liquid recording medium. U. S. Patent No. 4,243,994. [39] F. L. Cloutier D. K. Donald J. D. Meyer C. A. Tacklind Vaught, J. L. and H. H. Taub. Thermal ink jet printer. U. S. Patent No. 4,490,728, 1984. [40] A. M. Ga˜n´an-Calvo and J. M. Montanero. Revision of capillary cone-jet physics: Electrospray and flow focusing. Phys. Rev. E, 79:066305, 2009. [41] S. Goren and M. Gottlieb. Surface-tension-driven breakup of viscoelastic liquid threads. J . Fluid Mech., 120:245–266, 1982. [42] A.-C. Ruo, F. Chen, C.-A. Chung, and M.-H. Chang. Three-dimensional response of unrelaxed tension to instability of viscoelastic jets. J . Fluid Mech., 682:558–576, 2011.
128 BIBLIOGRAPHY [43] C. Clasen, J. Bico, V. M. Entov, and G. H. McKinley. ?gobbling drops?: the jetting-dripping transition in flows of polymer solutions. J. Fluid Mech, 636:5–40, 2009. [44] J. G. Oldroyd. On the formulation of rheological equations of state. Proc. Roy. Soc. Lond., 200:523–541, 1950. [45] D. F. James. Boger fluids. Annu. Rev. Fluid Mech., 41:129–142, 2009. [46] L.-J. Yang, M.-X. Tong, and Q.-F. Fu. Instability of viscoelastic annular liquid sheets subjected to unrelaxed axial elastic tension. J. Non-Newtonian Fluid Mech., 198:31–38, 2013. [47] P. Huerre and P. A. Monkewitz. Local and global instabilites in spatially developing flows. Annu. Rev. Fluid Mech., 22:473–537, 1990. [48] J. M. Montanero and A. M. Ga˜n´an-Calvo. Viscoelastic effects on the jettingdripping transition in co-flowing capillary jets. J. Fluid Mech., 610:249–260, 2008. [49] F. Li, A. M. Ga˜n´an-Calvo, J. M. L´opez-Herrera, X.-Y. Yin, and X.-Z. Yin. Absolute and convective instability of a charged viscoelastic liquid jet. J. Non-Newtonian Fluid Mech., 196:58–69, 2013. [50] S. Middlema. Stability of a viscoelastic jet. Chem. Eng. Sci., 20:1037–1040, 1965. [51] Z. Liu and Z. Liu. Linear analysis of three-dimensional instability of nonnewtonian liquid jets. J. Fluid Mech., 559:451–459, 2006.
BIBLIOGRAPHY 129 [52] M. Goldin, J. Yerushalmi, R. Pfeffer, and R. Shinnar. Breakup of a laminar capillary jet of a viscoelastic fluid. J. Fluid Mech., 38:689–711, 1969. [53] J. Eggers and E. Villermaux. Physics of liquid jets. Rep. Prog. Phys., 71:036601, 2008. [54] D. D. Joseph. Fluid dynamics of viscoelastic liquids. Springer-Verlag, 1990. [55] M. R. Khorrami, M. R. Malik, and R. L. Ash. Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys., 81:206–229, 1989. [56] R. J. Briggs. Electron-Stream Interaction with Plasmas. MIT Press, Cambridge, 1964. [57] M. R. Khorrami. Chebyshev spectral collocation method using a staggered grid for the stability of cylindrical flows. Int. J. Numer. Methods Fluids, 12:825–833, 1991. [58] M. Gaster. A note on the relation between temporally-increasing and spatiallyincreasing disturbances in hydrodynamic stability. J. Fluid Mech., 14:222–224, 1962. [59] S. J. Leib and M. E. Goldstein. Convective and absolute instability of a viscous liquid jet. Phys. Fluids, 29:952–954, 1986. [60] M. A. Herrada, J. M. Montanero, C. Ferrera, and A. M. Ga˜n´an-Calvo. Analysis of the dripping-jetting transition in compound capillary jets. J. Fluid Mech., 649:523–536, 2010.
130 BIBLIOGRAPHY [61] S. J. Gill and J. Gavis. Tensile stress in jets of viscoelastic fluids. I. J. Polym. Sci., 20:287–298, 1956. [62] P. Guillot, A. Colin, A. S. Utada, and A. Ajdari. Stability of a jet in confined pressure-driven biphasic flows at low Reynolds numbers. Phys. Rev. Lett., 99:104502, 2007. [63] T. Si, F. Li, X.-Y. Yin, and X.-Z. Yin. Modes in flow focusing and instability of coaxial liquid-gas jets. J. Fluid Mech., 629:1–23, 2009. [64] E. J. Vega, J. M. Montanero, M. A. Herrada, and A. M. Ga˜n´an-Calvo. Global and local instability of flow focusing: The influence of the geometry. Phys. Fluids, 22:064105, 2010. [65] L. Rayleigh. On the instability of jets. Proc. London Math. Soc., s1-10:4–13, 1878. [66] A. M. Ga˜n´an-Calvo, J. M. Montanero, L. Mart´ın-Banderas, and M. FloresMosquera. Building functional materials for health care and pharmacy from microfluidic principles and Flow Focusing. Adv. Drug Delivery Rev., 65:1447– 1469, 2013. [67] J. Rosell-Llompart and A. M. Ga˜n´an-Calvo. Turbulence in pneumatic flow focusing and flow blurring regimes. Phys. Rev. E, 77:036321, 2008. [68] M. Brinkmann and R. Lipowsky. Wetting morphologies on substrates with striped surface domains. J. Appl. Phys., 92:4296, 2002. [69] G. W. Young and S. H. Davis. Rivulet instabilities. J. Fluid Mech., 176:1–31, 1987.
BIBLIOGRAPHY 137 namic tip streaming and emission of charged drops from liquid cones. Nature Phys., 4:149–154, 2008. [122] Y. Kawana J. Kimura and T. Kuriyama. An immobilized enzyme membrane fabrication method using an ink jet nozzle. Biosensors, 4:41, 1988. [123] W. J. Lloyd and H. H. Taub. Ink jet printing. Output Hardcopy Devices, Chap. 13:p. 311, 1988. [124] G. A. Howell D. J. Hayes G. L. Bernardini, B. A. Rampy and C. J.Frederickson. Applications of piezoelectric fluid jetting devices to neuroscience research. J. Neurosci. Methods, 38:81, 1991. [125] T. M. Brennan. Method and apparatus for conducting an array of chemical reactions on a support surface. U.S. Patent No. 5,474,796, 1995. [126] O. Br D. Sziele. [127] H. P. Le. Progress and trends in ink-jet printing technology. J. Imaging Sci. Technol, 42:49, 1998. [128] F. L. Degertekin G. Per¸cin, A. Atalar and B. T. Khuri-Yakub. Micromachined two-dimensional array piezoelectrically actuated transducers. Appl. Phys, 72:1397, 1998. [129] Alvin U. Chen and Osman A. Basaran. A new method for significantly reducing drop radius without reducing nozzle radius in drop-on-demand drop production. Physics of fluids, 14:1, 2001.
138 BIBLIOGRAPHY [130] A. A. Castrej´on-Pita, J. R. Castrej´on-Pita, and G. D. Martin. A novel method to produce small droplets from large nozzles. Review of Scientific Instruments, 83:115105, 2012. [131] R. I. Issa. Solution of the implicitly discretised fluid flow equations by operatorsplitting. Journal of Computational Physics, 62:40–65, 1986. [132] B. V. Leer. Towards the ultimate conservative difference scheme. v - a secondorder sequel to godunov’s method. Journal of Computational Physics, 32:101– 136, 1979.